The oldest human character seen in Star Trek to have naturally aged (barring Flint) is McCoy, Kirk's near contemporary. If he hadn't fallen with that bridge, Kirk would presumably have a life expectancy similar to McCoy.
Did we ever really get much info about the TOS character's parents? Sarek was obviously alive for a while. We saw McCoy's dad die in TFF, at which point it was clear that he lived to be a really old age. I don't think we ever really heard much of anything regarding the rest of the casts' family.
TNG is another story. We had plenty of references and actually met a lot of their parents throughout the run of the show. Riker's dad, Troi's mom, both of Worf's parents...I don't remember if we ever met Geordi's mom, but I know she was mentioned in at least one episode.
We didn't get Kirk's parents info until 2009's Star Trek, and nothing really on any of the other TOS characters' parents (Chekov is presumably the son of an Andrei).
Captain Silva La Forge was a major focus of the episode "Interface". The episode also showed us Geordi's father and delved into his family life quite a bit. She was played by Madge Sinclair.
So, in canon, the parent count is this:
TOS
Kirk - Father: George (appears), Mother: Winona (appears)
Spock - Father: Sarek (appears), Mother: Amanda (appears)
McCoy - Father: David (appears)
Scott - ?
Uhura - ?
Sulu - ?
Chekov - Father: Andrei (implied)
Chapel - ?
Rand - ?
TAS
Arex - ?
M'Ress - ?
TNG
Picard - Father: Maurice (appears), Mother: Yvette (appears)
Riker - Father: Kyle (appears), Mother: Betty (mention)
La Forge - Father: Edward (appears), Mother: Silva (appears)
Yar - ?
Worf - Father: Mogh (mention), Mother: ? - Foster Father: Sergey (appears), Foster Mother: Helena (appears)
Beverly Crusher - Father: Paul (mention), Mother: Isabel (mention)
Troi - Father: Ian (appears), Mother: Lwaxana (appears)
Data - Father: Noonien (appears), Mother: Juliana (appears)
Wesley Crusher - Father: Jack (appears), Mother: Beverly (appears)
DS9
Sisko - Father: Joseph (appears), Mother: Sarah (appears) - Stepmother: ? (mention)
Odo - ?
Ezri Dax - Father: Lazo (mention), Mother: Yanas (appears)
Bashir - Father: Richard (appears), Mother: Amsha (appears)
Jadzia Dax - Father: Kela (mention)
Jake Sisko - Father: Benjamin (appears), Mother: Jennifer (appears)
O'Brien - Father: Michael (mention), Mother: ? (mention)
Quark - Father: Keldar (mention), Mother: Ishka (appears)
Kira - Father: Taban (appears), Mother: Meru (appears)
VOY
Janeway - Father: ? (appears)
Chakotay - Father: Kolopak (appears)
Torres - Father: John (appears), Mother: Miral (appears)
Kes - Father: Benaren (appears), Mother: Martis (appears)
Paris - Father: Owen (appears)
Neelix - ?
Doctor - Father: Lewis (appears)
Tuvok - Mother: T'Meni (mention)
Seven - Father: Magnus (appears), Mother: Erin (appears)
Kim - Father: John (appears), Mother: Mary (appears)
ENT
Archer - Father: Henry (appears), Mother: Sally (mention)
Phlox - ?
T'Pol - Mother: T'Les (appears)
Reed - Father: Stuart (appears), Mother: Mary (appears)
Mayweather - Father: ? (mention), Mother: Rianna (appears)
Sato - Father: ? (appears), Mother: ? (mention)
Tucker - Father: Charles (mention)
So, the Original Series (and Animated Series) have by far revealed the least of its main characters' parentage. But that was part of the overall vagueness to any background outside of the main three. And really outside of Spock, since we only learn about Kirk and McCoy's parentage in the movies.